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Header: I am really sorry for the confusing title. I am going to edit it so it fits the responses I get based on Mann Whitney U.

So I had this exact question on stack overflow about is the Mann Whitney U Test a test of medians or means. I really liked Glen_b's response, but it lead to a bit of new confusion about "pairwise" and the meaning of "rank sum". In particular here is what my notes say:

Mann-Whitney Test

Assumptions: Samples are independent from each other

And this nonparametric is for samples where the outcomes are not paired. Before we were assuming that both yi and zi were observed for the same data point. For example let's say we have the same virus [data that we did for the two sample paired Wilcoxian signed rank test]. Now we're just assuming that all the observations are independent of each other. This test is called the Mann-Whitney test. In the Mann-Whitney test we rank all of the yi and zi together. And then, add up the ranks of all the samples that come from the first set. All the ranks of the yi and the ranks of the samples in the second set, all the ranks of the zi. Whichever sum is smaller is compared again against the table that gives the significance of the difference."

What do people mean by "pairwise"? (See the link above for context) How can one have an unpaired sample test that uses "pairwise"?

What the heck is a median difference? Do we mean the median of the rank? Is the median difference a measure of central tendency or measure of location?

mlane
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  • So I thought I should post the original question I had that Glen_b managed to answer. (Part 1 of 4) I was studying some notes I had for my statistics class, and I was confused about the Mann Whitney U Test. I have heard that an "unpaired two sample median test" is a Mann Whitney U Test. However, the lecture notes I watch suggest that a Mann Whitney U Test is also called a Wilcoxian Signed Rank SUM Test. My confusion is that some articles say that it is a median test and some articles say that is not a median test, but a mean test. – mlane Jul 30 '19 at 22:03
  • (Part 2 of 4) This is an article that does say its a median test: https://stats.idre.ucla.edu/other/mult-pkg/faq/general/faq-why-is-the-mann-whitney-significant-when-the-medians-are-equal/ ""Consider the following example dataset of 120 observation (60 in each group) that has equal medians and a significant Mann-Whitney-Wilcoxon test."" – mlane Jul 30 '19 at 22:04
  • (Part 3 of 4) Another article: http://sphweb.bumc.bu.edu/otlt/mph-modules/bs/bs704_nonparametric/BS704_Nonparametric4.html ""A popular test to compare outcomes between two independent groups is the Mann Whitney U test. The Mann Whitney U test, sometimes called the Mann Whitney Wilcoxon Test or the Wilcoxon Rank Sum Test, is used to test whether two samples are likely to derive from the same population (i.e., that the two populations have the same shape). Some investigators interpret this test as comparing the medians between the two populations."" – mlane Jul 30 '19 at 22:10
  • (Part 4 of 4) While this says it is not the medians but the means: https://www.graphpad.com/guides/prism/7/statistics/index.htm?stat_nonparametric_tests_dont_compa.htm ""The Mann-Whitney test compares the mean ranks -- it does not compare medians and does not compare distributions. More generally, the P value answers this question: What is the chance that a randomly selected value from the population with the larger mean rank is greater than a randomly selected value from the other population?"" So is Mann-Whitney a comparison of means or medians? If its means is it really non-parametric? – mlane Jul 30 '19 at 22:24
  • Glen clarified his original post, I think you are sorted now? :) Also check the original publication [here](https://pdfs.semanticscholar.org/60f7/88264e6278374be9dabd8e2a644cc65129b6.pdf) it says nothing about means or medians. It concerns medians only if we assume that the alternative hypothesis is restricted to a shift in location. (Which we implicitly do in many cases but that's not what it says on the (test's) tin!) – usεr11852 Jul 30 '19 at 23:03
  • @usεr11852 omg he did? Wow! Glen noticed. What should I do now? Should I answer my own question or close my own post? – mlane Jul 30 '19 at 23:26
  • We are good sports here; if ones makes a comment and usually the OP takes notice. Feel free to answer your own question if you think you want to express your understanding of the issue. You clearly have done some reading on this. – usεr11852 Jul 30 '19 at 23:35
  • Sorry, I didn't even see this question until now. I edited the other answer because (as usεr11852 pointed out in comments there, though the comment is now deleted) it was ambiguous. If there's anything not already covered elsewhere, please rephrase this to focus on just whatever remains to be explained. In relation to your title, Mann-Whitney is not a test of medians, so the opening premise is false. You may want to address that first; I am happy to clarify, though it is discussed in other posts so a search might cover what you need. – Glen_b Jul 31 '19 at 01:05
  • I don't think you should wait to get answers in order to figure out what your question is (per the opening of your question) -- that's backwards; it may end up leaving answerers trying to chase a moving target. Just ask the clearest question you can about whatever you don't currently understand; you can choose to answer it once the question is clarified if you wish. Currently the question is a bit of a muddle. I plan to put it on hold until it is in a more suitable form; you should edit it (&while not usual practice, in this case you can ping me in comments for attention once you do) – Glen_b Jul 31 '19 at 01:09
  • Actually there's enough confusions in your reported understanding in comments that perhaps we had better address some of those (I can't do it right now though). I'll leave the question open for the moment but please edit the question to clarify what you need the most. – Glen_b Jul 31 '19 at 02:36
  • I plan on doing that either today or tommorow @Glen_b I will message you when I am done. :) – mlane Jul 31 '19 at 22:15

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